Ward’s Hierarchical Agglomerative Clustering Method: Which Algorithms Implement Ward’s Criterion?

This paper investigates Ward's error sum of squares hierarchical clustering method, which has been very widely used since its first description by Ward in a 1963 publication and has been generalized in various ways. The authors observe that two different algorithms appear in the literature and in software, both announcing that they implement the Ward clustering method. However, when these two algorithms are applied to the same distance matrix, they produce different results: one algorithm preserves Ward's original criterion while the other does not.

Through survey work and case studies, the paper clarifies this discrepancy and identifies which algorithm actually corresponds to Ward's criterion. The authors emphasize that this clarification is useful for everyone involved in developing software for data analysis using Ward's hierarchical clustering method, helping practitioners avoid inadvertently using a variant that does not implement the intended criterion and thereby ensuring more consistent and correct clustering results.